Annales Academire Scientiarum FennicreSeries A. I. MathematicaVolumen 10, 1985, 4ll-417

Commentationes in honoremOlli Lehto

LX annos nato

ON THE STABILITY OF IDENTIFICATION PATTERNSFOR DIRICHLET REGIONS

MARJATTA NÄÄTÄNBN

1. Introduction

Let G be a purely hyperbolic Fuchsian group of the first kind with genus g>1and let D(z) be the Dirichlet region for G with center z. It was proved by Beardon

[2, Theorem 9.4.5] that for a.e. z D(z) has all cycles of length 3 and, hence,129-6 sides. Let p(g) give the number of possible identification patterns. In [3]we showed tbat p(2):8. For i:1,...,p(g) let

O,: {zlD(z) has 129-6 sides and pattern i}.

It is obvious that the sets D; are invariant under G. Here it is shown that thesets D; are open and properties for their boundary curves are proved. Chapters 3

and 4 deal with the case g-2. It is shown that the group of the regular octagon,introduced in [4, Theorem 5.1], only has pattern 6 and its degenerates as patternsfor D(z). Groups which change their pattern according to elementary moves [3,Chapter 5l are constructed in Chapter 4.

I am indebted to S. Mustonen, L Mellin and H. Haario for help with computerwork, to S. Rokka for the figures, and to M. Lehtinen for valuable comments.

2. The sets D; nd \Dt

In [2, Theorem 9.4.5] it is shown that if D(z)has less than 129-6 sides, thenthere exist distinct ,€GV, i:1,2,3, such that

(2.t) lmfz,flQ), f2Q),fs(z)l: 0,

where the cross-ratio cannot be constant. Since G has a countable number oftrans-formations, the "exceptional" centers lie on a countable number of curves definedby (2.1). Each curve is algebraic, since (2.1) gives a polynomial of degree 8 in two realvariables.

Definition 2.I. Let r€GV bedistinct, i:1,2,3, andlet y be the algebraiccurae corresponding to the condition Q.l). IAe defi.ne f as the set of all curues y.

4t2 Mlnrarra NÄÄIÄNBN

Remark 2.1. By the invariance of the cross-ratio

{z l[2, .f1 Q), f2 Q), f, (z)]( R] : {z lfh (z), hf, (t), hf, (z), hf" (4]$]for each Möbius transformation å. Hence the set i- could also be defined by replacingconditions of type (2.1) by conditions

Imlgr(z), gz(z), g"(z), gn@)l: 0,

where g;€G are distinct, i:1,2,3, 4.

Theorem 2.1. Each cycle of length 4 in D(z) corresponds to z lying on acurue yQl.

Proof. Let P be a vertex of D(z) with the other vertices in its cycle

Sr(P),gzgJP),EsCzgJP). Then z,grr!), krgr\-t(z) and (gsgggr)-1(z) are con-cyclic with center P. Because g. is a Möbius transformation, this is equivalent tothe points Et(z), z, SrtQ) and (gsgz)-r(z) being concyclic with center gr(P). Thealgebraic curves given by

Im l.r, gl'(t), (grgr.)-t(r), (grgrgr)-t(r)l : 0and

Im [2, g;t (z), (9" gr)-' (z), gt(4) : 0

are the same. A simjlar argument applies to the other vertices in the cycle of P.

Theorem 2.2. ff D(z) has one cycle of length 4 and the others of length 3,

then in a neighbourhood U of t,Ullul \Din(J:(J ny, where y€l, andfor w(U nythe identification pattern of D(w) is the same as that of D(z), while when w crosses

7 rn U\{z} a side-pairing tansformation is changed in D(w).

Proof. If P is a vertex with cycle length 3, then by continuity and discretenessz has a neighbourhood in which D (w) has a vertex in a neighbourhood of P with thesame cycle length and adjacent sides labelled by the same mappings as P has forD(z').

The condition for Q being a vertex of D(z\ with cycle Iength 4 and with adjacentsides labelled by fr,f", the sidefr being degenerate at Q, is, in hyperbolic distances,

a(Q, z): Q(Q,tLQ)): e(Q,.fr(r)): o(Q,f"(4) = a(Q, ce))for all g€G\9,,(,.fr,fr\ ud f2Q\ lies on the circle with center p on the partbetween fr(z) and fr(z) not containing z. Then also the side fllf, is degenerate.Because ofcontinuity and discretenessthereexistsaneighbourhood U ofz such thatfor w(U ny, where 7 is given by (2.1), the pattern of D(w) is the same as that ofD(z),whtle for w€U\t, D(w)has 129-6 sides since all cycles are of length 3.

We claim the side-pairing mapping added to D(z) is changed between f2 and

f;'f" us the curve 7 is crossed in U\{z}. Denote R(w):lw,fr(w),fr(w),f"(w)1.Then R is not a constant. Let g- be the Möbius transformation which maps r,rl,

fr(r),.f"(w) to 0, 1,6, respectively. Then R(w):g.(fr(w)). Let C(w)be the open

Stability of identification patterns for Dirichlet regions

disc with w,fr(w),fg(w) on åC(p). Then s*(0C1w11:R and s-(C(r)) is either

the upper or lower half-plane. Assume it is the upper one. Then, if ImÄ(w)>0,it follows that fz(w)ec(w) and the side labelled by f^ appears in D(w) between

thesideslabelledbyfrandfu'IfImR(w)=0,thenfr(w)isnotintheclosureofC(w) and there is no side between the sides fi and fg in D(w). Since Ä is a rational

function, there is a neighbourhood N of z such that Ä(1/) covers a neighbourhood ofR(z) at least once and for each w€N\{z} Ä is a homeomorphism when restricted

to a neighbourhood N' of w.If w€y\{z}, then R(w)€R.Denote (Ir:{w(N'l ImR(w)=O}, (Iz:{w€N'lIm Ä(w)<0}. Then, when

wC.(Jt, D(w) has a side labelledby f, and, when w€.[Jz,S is not in D(w), and the

added side is labelled by .fr'.f".The common boundary of Ut and [/2 in N' is 7,

hence when lv crosses y a side-pairing mapping is changed in D(w).

Theorem 2.3. Let t<Ui:? \Di and let I be the number of the cyc.lgs in

D(z) hauing length ry,>4, k:1, ..,,1. Then in a neighbourhood (l o7 z,ll!! \Dt

Iies on at most }I=r("t) curoes y€f. Fo,r each !,T^U is the union of finitely

many arcs which intersect only at z, edch homeomorphic to an open interual of real

numbers.

Proof. For k: l, ..., I let Po be a vertex of D(z) with length of the cycle no>4.

Then there are exactly no points of the set G(z) on a circle wiih center Po. Each choice

of 4 of them determines a curve 7 through z by the condition

(2.2) Im [g, (r), gr(z), gr(z), Snk)] - 0.

As in Theorem2.l, by mapping Po to another vertex in the cycle, we see that

each vertex in the same cycle gives rise to the same curves 7€i-.Each curve y is the preimage of R under the non-constant rational mapping

R of Theorem 2.2. Hence the last assertion holds in a suitably chosen neighbourhood

U of z(y.

Remark 2.2. Let CID(0) denote the closure of D(0). It sufrces to consider

points in CID(O) to investigate the sets Dr, i:1,...,p(g), for a group G, since

the sets Di are invariant under G. By Theorem 2.3 and the compactness of CID(O),

Ui!? \o,nClD(0) is on a finite number of curves y€f.

Theorem 2.4, Let the group G act on the open unit disc U, let z(D;,i:1,...,pG), qnd let B be the union of subsets of U fulfiiling conditions of type

(2.1), where fr,fr,f" label adjacent sides of D(z). Let 14/ be the component ofU\B containing z. Then WcD, and |lAc.DDi.

Proof. Since z(D;, i:1,...,p(g), D(z) has a maximal number of sides.

Hence, because of continuity,for w€W, D(w) has the same pattern as D(z), since

the pattern of D(w) can change from that of D(z) only if some of the sides of D(z)

4t3

4t4 MaRrnrrn NÄÄrÄNnN

degenerate to vertices, i.e., if the center w fulflls at least one condition of type (2.1)

for mappings labelling adjacent sides of D(z). Hence WcD;.For w(\W, a condition of type (2.1) is valid for mappings labelling three adja-

cent sides of D(z) and, hence, at least one side-pair of D(z) is degenerate in D(w).Thus |Wc.|Di.

Corollary 2.1. The set Di, i:|, ...,p(g\, is open.

3. Application to the group of the regular octagon

In [4] we studied the group with D(0) the regular octagon with diametricallyopposite pairings. We now derive the curves l€.f of Theorem 2.3 through 0. In

this case Utrål, fies on at most (i):^ curves in a neighbourhood of 0. In

Theorem 3.1, the dependence of D(z) on z is studied for this group.The generators are fr,fr,.fr,.fn with the relation frfrfrfnfrt.fr'f;'f;t:I

and f1ra1: g-o frgo, k: 1,2, 3, where g(z\:(exp (isnfi)) z.

The cycle of a vertex of D(0) has length S and UtråD; is in a neighbourhoodof 0 on the curves y obtained from mappin5s l,.fr, fr.fr, frfrfu,.frfr.f"fu, fnfrfr,.fnfu, fn, labelling adjacent Dirichlet regions. Since the mappings are Möbius trans-formations and the generators are conjugates, we obtain for example: if we denotew:g-'(z), then lm[z,ft(z),frfr(z\,ftfrfr(z)l:0 is equivalent to Im [fi(w),ftfr(r),ftfrf"(w),frfrf"fn(w)l:0. Hence the set of points satisfying the latterequation is obtained by rotating by g-t the set of points satisfying the former equa-tion. Hence it suffices to consider the cyclic choices of four of the mappings labellingthe Dirichlet regions around P and to take rotations by the powers of g of the al-gebraic curves we have obtained to get all curves through 0 on which l)?=r0O,can lie in a neighbourhood of 0. These are shown in Figure I, calculateil by a com-puter.

Theorem 3.1. The group of the regular octdgon only has pattern 6l3l and itsdegenerates as patterns for D(z).

Proof. Step l. To get the patterns for D(z), z€ClD(O), it suffices to considerthe sector with angle ln,Snl4l in the origin: if wedenote fi*n:f-', i:1,...,4,then fr:g-%go holds for i,j,k:1,...,8, j+k:i (mod 8). Choose i:1,...,7.We claim that the pattern for D(z'),where z':gt(z), is obtained from D(z) byreplacing the generator fi,i:I,...,4, by fi-, (mod 8). This follows from thefact that g' is a Möbius transformation and grfiQ):ti.fig-i (z'):fr-i(z').

Step 2. Let r and A be the radii of the inscribed and circumscribed circles forD(0). We claim it suffices to consider z rn the triangle Z(Figure 2\ with vertices in

Stability of identification patterns for Dirichlet regions 4t5

the hyperbolic polar coordinates 0, Q:(r,n), 5:(r,5nl4). Let P:(R,9nl8)'TheselD(P) isaregularoctagonwithdiagonalpairingsbl fnfrfr,fr.frf",frfrfnt,frf;'f;t. Let hbe the reflection in the line through Q and S. Then å maps D(P) to

liöl"ta h-Lfrh:(ftnftrfr)-',h-'frh:f'fufr',h-tf"h:7rftrf;t,h-'fnh:Ut.frfr\-t'We deduce that D(h(z)) is obtained from D(z) by replacing each fi' i:1, "',4,by its conjugate (and reversing the order). Also, the reflections of the curves in Zindicating exceptional centers are curves indicating exceptional centers n hQ)belongipg to the conjugate mappings.

step 3. we finish by calculating D (z) for z QT. By continuity, in a neighbourhood

of 0, D(z) only has sides labelled by g, where a(g(OD is adjacent to D(0). We want

to find a point zo#0 in this neighbourhood and calculate D(zo).

Figure 1

416 Manrarru NÄÄrÄNpN

The set D(z) is in a neighbourhood of 0 obtained from D(0) so that some of thesides degenerated to points open up. The obtained pattern can only change if someof the sides of D(z) degenerate to points. Let gr, gz, gs be adjacent sides of D(z).The side 92 degenerates to a point on the curve Imlz, gr(z), gzQ), gs!)l:0. Thereare two cases to consider depending on whether 92 labels a side or a "degenerateside" of D(0).

Case L The regions D(gr(O), D(g'r(0)), D(0) and, respectivety, D(sz(O),D(gr(0)), D(0) have a common vertex, the vertices are adjacent, and, gr labels theside of D(0) between them. The corresponding curves are obtained by rotation bypowers of g from the following special case: Take;f, for gr. There are six choices forgt and six for gs. The number of curves to be drawn by a computer can be diminishedby means of symmetry. All obtained curves have euclidean distance at least 0.4from 0.

Case 2. The regions o(gr(0)), D(cr(O)), a(gr(0D and D(0) have a commonvertex. The corresponding curves are in Figure 1.

Hence for D(-0.3, -0.04) it suffices to consider the mappings tabelling Di-richlet regions adjacent to D(0); it will have 18 sides. The sides are: fr, frff,r,(fnf"fr)-', "fr', (.frf)-t, fg, faL, frt, .fr, (fnf")-t, .fr', (fr.frf")-., fEftL,.fa,

-fnf", fnf"f^, frfrf", frfr. The pattern is 6. Aside, saylf , disappears by becominga vertex in D(z) if Im[z,frfr(z),f,,(t),frfn t(z)]:Q. The curves for all sides arealready calculated and the result is that the pattern above is preserved in z, withthe sides fn.f", fr.f^.f" degenerating on 0Q, .fr.fr, fnf"f, on 0S, fz, fs on eS andfr.fn t at O, f, at S, fa at Q.

The conclusion is that the group of the regular octagon has for patterns forD(z), z€ClD(0), pattern 6, its degenerates with 14 sides on the lines drawn inFigure 2 (and their rotations by powers of g), and the regular octagon with diagonalidentifications at 0, at the vertices of D(0) and at the midpoints of sides of D(0).The patterns for z, z(ClD("f(0)), where/is in the group, are obtained by conjugation.

Figure 2

Stability of identification patterns for Dirichlet regions 4t7

Remark 3.1. Figure 2 is obtained by using a computer, but exact calculations

can also be made. For example, the cross-ratio lz,fnQ),.fn.ft(z),fnf"fr(z)l canberepresented as a quotient of two polynomials of degree fourin z,withreal coefficients.

Hence the side fafgdegenerates on the real axis.

4. Examples of groups with at least two patterns

We construct a group which has points in D. and Dr. Other groups corresponding

to the transition pattern [3, Chapter 5] can be constructed in a similar way.

The group is obtained by constructing D(0): Let P be a sixteengon with every

fourth vertex at an equal distance from 0 and with angles nl2, all others at an equal

distance from 0 and with angles 2nf3, constructed by using [], Theorem 1]. Then

the angle bisectors are concurrent at 0 and the sides ofP can be identified by using

the third sequence in [3, Chapter 6]. Because of the conditions used, P is D(0) forthe group generated by its side-pairing transformations.

The point 0 is on the curve 7€I where the cross-ratio corresponding to the

cycle of length 4 is real, and it follows from Theorem 2.2 that there exists a neigh-

bourhood N of 0 such that in each component of ,l/\ one of the degenerated side-

.pairs of D(0) opens up and the pair opening up is changed when crossing 7. The

change ofthe sides changes the pattern between I and 2.

References

[l] BrnnooN, A. F.: Hyperbolic polygons and Fuchsian gxoups. - J. London Math. Soc. (2) 20,

1979,247---254.

[2] BrenooN, A. F.: The geometry of discrete groups. - Graduate Texts in Mathematics 9l..Springer-Verlag, New York, 1983.

[3] JoncnrsrN, T., and M. NliÄrÄrm: Surfaces of genus 2: Generic fundamental polygons. - Quart.J. Math. Oxford Ser. (2) 33, 1982, 451-461.

[4] NiiÄrÄNrN, M.: Regular n-gons and Fuchsian groups. - Ann. Acad. Sci. Fenn' Ser' A I Math' 7'

1982, 291-300.

University of HelsinkiDepartment of MathematicsSF-00100 HelsinkiFinland

Received 19 March 1984